In the present days the technological advances permit the study of the history and evolution of the Universe in a more profound manner, leading to a series of questions and breakthroughs from a scientific point of view. The exploration of various questions related to the history of the Universe can open new extraordinary possibilities in the long run. On a short time scale, the quest for the major cosmological scientific questions pose new specific difficulties to various technological areas, leading to the development of new technologies in applied science. The discovery of the accelerated expansion of the current Universe was a breakthrough at the end of the last millennium [1]}, configuring new specific areas in physics. This new curious phenomenon has been probed through various phenomenological studies [2]}, [3]}, representing a viable and intriguing aspect of modern physics [4]}. The simplest dark energy model is represented by the \(\Lambda \) CDM scenario where the accelerated expansion is driven by a cosmological constant [5]}. However, this model has various limitations and cannot explain various observed features, including the late time dynamical evolution of the dark energy equation of state [5]}. In order to overcome these issues, a new theoretical direction has emerged, the extended theories of gravity [7]}. These theories are based on the mathematical approach of basic general relativity, extending the fundamental action to a more general or special form [8]}, [9]}. It can be seen that the modified gravity theories can explain various aspects from the evolution of the Universe [10]}. In these theories, the fundamental action is extended, being replaced with a generic functional which takes into account possible influences from various invariant geometrical quantities [11]}. Another viable approach consists in adding one or more scalar fields in the fundamental action [12]}, [13]}, relating the evolution of the late time acceleration of the Universe with the dynamics induced by the scalar field(s). The scalar fields which can trigger the acceleration of the Universe can be in the form of quintessence models, with a canonical kinetic energy [14]}, [15]}, embedding also a potential term. Another approach where the scalar fields are non-canonical is represented by phantom dark energy models, an approach which can explain the super–acceleration [16]}, [17]}, [18]}, [19]}. Soon after the emergence of quintessence and phantom models, a new intriguing aspect was discovered – the crossing of the cosmological constant boundary in the recent past for the dark energy equation of state [20]}, [21]}, [22]}. This phenomenon triggered the formation of quintom models [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, a complex construction which includes two scalar fields, one with a negative kinetic energy, and another, with a canonical kinetic energy. Such a construction can explain the crossing of the phantom divide line in the recent past, being extended also in the non–minimal case [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}.
